A Multi-scale Approach to 3D Scattered Data Interpolation with Compactly Supported Basis Functions

Transcription

1 Shape Modeling International 2003 Seoul, Korea A Multi-scale Approach to 3D Scattered Data Interpolation with Compactly Supported Basis Functions Yutaa Ohtae Alexander Belyaev Hans-Peter Seidel

2 Objective given scattered points interpolate them by f ( x, y, z) = 0 Interpolating

3 Implicit Representation Surface: f(x,y,z)=0 (implicit surface) Inside: f(x,y,z)>0 Outside: f(x,y,z)<0 Zero-level set of w=f(x,y,z) A polygonization of f(x,y,z)=0

4 Advantages of Implicits Constructive Solid Geometry Union, intersection, difference, blending, embossing, / = U blending =

5 Advantages of Implicits Reconstruction of missing parts of digitized objects Zero-level set of f(x,y,z) represents a closed surface.

6 Previous Wors Using Radial Basis Functions (RBFs) Murai et al Blobby model Savcheno et al. 1995, Tur et al Thin-plate RBF splines Morse et al Compactly supported piecewise polynomial RBFs Carr et al Biharmonic RBF splines and truncated series expansions Can process large point sets

7 Interpolating with Compactly Supported RBFs Fast (solving a sparse linear system) but regular sampling is required no ability to reconstruct missed data does not define a solid irregular sampling causes a problem not a solid No ability to reconstruct missed data

8 Our Approach Multi-scale approach few Points many large Support size small

9 Contents Single-scale Interpolation Local shape function x RBF Multi-scale Interpolation Results and Remaining Problems

10 Standard RBF Interpolations

11 Standard RBF Interpolations f ( x) = 0 On-surface point

12 Standard RBF Interpolations Off-surface point On-surface point f ( x) = 0

13 Standard RBF Interpolations f + 1 ( x) 0 Off-surface point = 0 p i on -/off - surface points f On-surface point λ φ ( x p ) + i i ( p ) = v ( i 1,2,..., N i i = f ( x) = 0 ) l( x) System of linear equations with unnown λ i

14 Basic Idea of Interpolation 1. Define local shape functions (in implicit form) 2. Blend the functions (weighted sum) Solving a sparse linear system.

15 Basic Idea of Interpolation 1. Define local shape functions (in implicit form) 2. Blend the functions (weighted sum) Solving a sparse linear system.

16 Basic Idea of Interpolation 1. Define local shape functions (in implicit form) 2. Blend the functions (weighted sum) Solving a sparse linear system.

17 Basic Idea of Interpolation 1. Define local shape functions (in implicit form) 2. Blend the functions (weighted sum) Solving a sparse linear system.

18 Basic Idea of Interpolation 1. Define local shape functions (in implicit form) 2. Blend the functions (weighted sum) Solving a sparse linear system.

19 Basic Idea of Interpolation 1. Define local shape functions (in implicit form) 2. Blend the functions (weighted sum) Solving a sparse linear system.

20 Basic Idea of Interpolation 1. Define local shape functions (in implicit form) 2. Blend the functions (weighted sum) Solving a sparse linear system.

21 Formulation ( g ( ) + λ ) i i f ( x) = x ( x p ) p i P Local shape function in implicit form φσ Unnown (Shift amount) i Compactly supported radial basis (blending) function 2D D Graph of ( x) φ( x) g i φ( r) 4 (1 r) (4r + 1) if r < 1 = 0 else Introduced by Wendland 1995

22 Results of single-level level interpolation 35K K points 5 sec. 134K K points 47 sec. Holes remain Narrow band support

23 Results for Irregular Sampling Irregularly sampled points Many holes remain because of small support of basis functions, but choosing a large support leads to an inefficient computation procedure.

24 Contents Single-scale Interpolation Multi-scale Interpolation Results and Remaining Problems

25 Algorithm 1. Construction of a point-set hierarchy. 2. Coarse-to-fine interpolations.

26 Construction of Point Hierarchy Uniform octree-based down-sampling. Coordinates and normals are the average of leaf nodes. Final level is decided according to density of points. Appended to hierarchy Level 1 (2 3 cells) Level 2 Level 3 Level 4 Level 5 Level 6 Given points

27 Coarse-to to-fine interpolation Level -1 f 1 ( x) = 0 Level f ( x) = 0 + o (x)

28 Coarse-to to-fine interpolation Level -1 f 1 ( x) = 0 Level f ( x) = 0 + o (x) f (x) = f 1 ( x) + o ( x), f 0 ( x) = 1

29 Coarse-to to-fine interpolation Level -1 f 1 ( x) = 0 Level f ( x) = 0 + o (x) f (x) = f 1 ( x) + o ( x), f 0 ( x) = 1 Same form f (x) as in the single scale ( + ) g i ( x) λi o ( x) = ( x p ) p i P φσ i

30 Coarse-to to-fine interpolation Level -1 f 1 ( x) = 0 Level f ( x) = 0 + o (x) f (x) = f 1 ( x) + o ( x), f 0 ( x) = 1 Same form f (x) as in the single scale ( + ) g i ( x) λi o ( x) = ( x p ) p i P φσ σ i = σ 1 / 2, σ 1 Diameter of object = 0.75L

31 Contents Single-scale Interpolation Multi-scale Interpolation Results and Remaining Problems

32 19 min. 332Mbyte Pentium GHz 7.5 min. 198Mbyte Level 9 (final level) Level 8 544K K points Approximation (error < 2-8 ) 901K functions 363 K functions

33 Comparison with method by Carr[SIG01] (FastRBF( FastRBF) Original 13K points FastRBF 30 sec. Our method 7 sec.

34 Noise comes from noisy boundary Points with normals from a merged mesh by VRIP (stand scans only)

35 Irregular Sampling Data 90% decimated Smooth joint

36 Feature Based Shape Reconstruction Inter- polation Features (ridges and ravines) Only feature points are ept Reconstruction result

37 Points with normals from mesh Points with noisy normals Polygonization f=0

38 Complicated Topological Object Point set surface Level1 Level2 Level3 Level4 Level5 Level6

39 Extra zero-level sets If the object has very thin parts, extra zero-level sets may appear. Octree based down-sampling is not sensitive for topological changes. A smart down-sampling procedure is required. No extra zero-level set inside the bounding box Extra zero-level sets appear near thin parts.

40 Sharp Features Original mesh with sharp features FastRBF (bi-harmonic) The proposed method

41 Shape Textures From two bunny s range images Holes are filled, but Too smooth

42 Summary Multi-scale approach to CS-RBFs Simple and fast Robust with respect to Irregular sampling Quality of normals Future Wor Avoiding extra zero-level sets Sharp features Shape texture reconstruction

43 Conclusion Multi-scale Interpolation with CS-RBFs Seems to be promising => to be continued